# RD Sharma Solutions For Class 7 Maths Chapter 16 Congruence Exercise 16.2

RD Sharma Solutions for Class 7 Maths Exercise 16.2 of Chapter 16 Congruence in PDF are available here. Students can refer and download it from the available links. This exercise has ten main questions along with many sub-questions. RD Sharma Solutions for Class 7 provides solutions for all chapters covered in the textbook. This exercise deals with the congruence of two triangles and sufficient condition for congruence of two triangles. Let us have a look at some of the important topics present in this exercise.

• Congruence of two triangles
• Sufficient condition for congruence of two triangles
• The side-side-side congruence condition

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Exercise 16.2 Page No: 16.8

1. In the following pairs of triangle (Fig. 12 to 15), the lengths of the sides are indicated along sides. By applying SSS condition, determine which are congruent. State the result in symbolic form.

Solution:

(i) InÂ Î”Â ABC andÂ Î”Â DEF

AB = DE = 4.5 cm (Side)

BC = EF = 6 cm (Side) and

AC = DF = 4 cm (Side)

SSS criterion is two triangles are congruent, if the three sides of triangle are respectively equal to the three sides of the other triangle.

Therefore, by SSS criterion of congruence,Â Î”ABCÂ â‰…Â Î”DEF

(ii) InÂ Î”Â ACB andÂ Î”Â ADB

AC = AD = 5.5cm (Side)

BC = BD = 5cm (Side) and

AB = AB = 6cm (Side)

SSS criterion is two triangles are congruent, if the three sides of triangle are respectively equal to the three sides of the other triangle.

Therefore, by SSS criterion of congruence,Â Î”ACBÂ â‰…Â Î”ADB

(iii) InÂ Î”Â ABD andÂ Î”Â FEC,

AB = FE = 5cm (Side)

AD = FC = 10.5cm (Side)

BD = CE = 7cm (Side)

SSS criterion is two triangles are congruent, if the three sides of triangle are respectively equal to the three sides of the other triangle.

Therefore, by SSS criterion of congruence,Â Î”ABDÂ â‰…Â Î”FEC

(iv) InÂ Î”Â ABO andÂ Î”Â DOC,

AB = DC = 4cm (Side)

AO = OC = 2cm (Side)

BO = OD = 3.5cm (Side)

SSS criterion is two triangles are congruent, if the three sides of triangle are respectively equal to the three sides of the other triangle.

Therefore, by SSS criterion of congruence,Â Î”ABOÂ â‰…Â Î”ODC

2. In fig.16, AD = DC and AB = BC

(i) IsÂ Î”ABDÂ â‰…Â Î”CBD?

(ii) State the three parts of matching pairs you have used to answer (i).

Solution:

(i) YesÂ Î”ABD â‰…Î”CBD by the SSS criterion.

SSS criterion is two triangles are congruent, if the three sides of triangle are respectively equal to the three sides of the other triangle.

Hence Î”ABD â‰…Î”CBD

(ii) We have used the three conditions in the SSS criterion as follows:

AD = DC

AB = BC and

DB = BD

3. In Fig. 17, AB = DC and BC = AD.

(i) IsÂ Î”ABCÂ â‰…Â Î”CDA?

(ii) What congruence condition have you used?

(iii) You have used some fact, not given in the question, what is that?

Solution:

(i) From the figure we have AB = DC

BC = AD

And AC = CA

SSS criterion is two triangles are congruent, if the three sides of triangle are respectively equal to the three sides of the other triangle.

Therefore by SSS criterionÂ Î”ABCÂ â‰…Â Î”CDA

(ii) We have used Side Side Side congruence condition with one side common in both the triangles.

(iii)Yes, we have used the fact that AC = CA.

4. InÂ Î”PQRÂ â‰…Â Î”EFD,

(i) Which side ofÂ Î”PQRÂ equals ED?

(ii) Which angle ofÂ Î”PQRÂ equals angle E?

Solution:

(i) PR = ED

Since the corresponding sides of congruent triangles are equal.

(ii)Â âˆ QPR =Â âˆ FED

Since the corresponding angles of congruent triangles are equal.

5. Triangles ABC and PQR are both isosceles with AB = AC and PQ = PR respectively. If also, AB = PQ and BC = QR, are the two triangles congruent? Which condition do you use?

It âˆ B = 50Â°, what is the measure of âˆ R?

Solution:

Given that AB = AC in isoscelesÂ Î”ABC

And PQ = PR in isoscelesÂ Î”PQR.

Also given that AB = PQ and QR = BC.

Therefore, AC = PR (AB = AC, PQ = PR and AB = PQ)

Hence,Â Î”ABCÂ â‰…Â Î”PQR

Now

âˆ ABC =Â âˆ PQR (Since triangles are congruent)

However,Â Î”PQR is isosceles.

Therefore,Â âˆ PRQ =Â âˆ PQR =Â âˆ ABC = 50o

6. ABC and DBC are both isosceles triangles on a common base BC such that A and D lie on the same side of BC. Are triangles ADB and ADC congruent? Which condition do you use? If âˆ BAC = 40Â° andÂ âˆ BDC = 100Â°, then findÂ âˆ ADB.

Solution:

Given ABC and DBC are both isosceles triangles on a common base BC

âˆ BAD =Â âˆ CAD (corresponding parts of congruent triangles)

âˆ BAD +Â âˆ CAD = 40o/ 2

âˆ BAD = 40o/2 =20o

âˆ ABC +Â âˆ BCA +Â âˆ BAC = 180o (Angle sum property)

SinceÂ Î”ABC is an isosceles triangle,

âˆ ABC =Â âˆ BCA

âˆ ABC +âˆ ABC + 40o = 180o

2Â âˆ ABC = 180oâ€“ 40o = 140o

âˆ ABC = 140o/2 = 70o

âˆ DBC +Â âˆ Â BCD +Â âˆ Â BDC = 180o (Angle sum property)

SinceÂ Î”DBC is an isosceles triangle,Â âˆ Â DBC =Â âˆ BCD

âˆ DBC +Â âˆ DBC + 100oÂ = 180o

2Â âˆ DBC = 180Â°â€“ 100oÂ = 80o

âˆ DBC = 80o/2 = 40o

InÂ Î”Â BAD,

âˆ ABD +Â âˆ BAD +Â âˆ ADB = 180o (Angle sum property)

30o + 20o +Â âˆ ADB = 180o (âˆ ADB =Â âˆ ABC â€“Â âˆ DBC),

âˆ ADB = 180o– 20oâ€“ 30o

âˆ ADB = 130o

7. Î”Â ABC andÂ Î”ABD are on a common base AB, and AC = BD and BC = AD as shown in Fig. 18. Which of the following statements is true?

(i)Â Î”ABCÂ â‰…Â Î”ABD

(ii)Â Î”ABCÂ â‰…Â Î”ADB

(iii)Â Î”ABCÂ â‰…Â Î”BAD

Solution:

InÂ Î”ABC andÂ Î”BAD we have,

AC = BD (given)

BC = AD (given)

And AB = BA (corresponding parts of congruent triangles)

Therefore by SSS criterion of congruency, Î”ABCÂ â‰…Â Î”BAD

Therefore option (iii) is true.

8. In Fig. 19,Â Î”ABC is isosceles with AB = AC, D is the mid-point of base BC.

(i) IsÂ Î”ADBÂ â‰…Â Î”ADC?

(ii) State the three pairs of matching parts you use to arrive at your answer.

Solution:

(i) Given that AB = AC.

Also since D is the midpoint of BC, BD = DC

Also, AD = DA

Therefore by SSS condition,

Î”ADBÂ â‰…Â Î”ADC

(ii)We have used AB, AC; BD, DC and AD, DA

9. In fig. 20,Â Î”ABC is isosceles with AB = AC. State ifÂ Î”ABCÂ â‰…Â Î”ACB. If yes, state three relations that you use to arrive at your answer.

Solution:

Given that Î”ABC is isosceles with AB = AC

SSS criterion is two triangles are congruent, if the three sides of triangle are respectively equal to the three sides of the other triangle.

Î”ABCÂ â‰…Â Î”ACBby SSS condition.

Since, ABC is an isosceles triangle, AB = AC and BC = CB

10. Triangles ABC and DBC have side BC common, AB = BD and AC = CD. Are the two triangles congruent? State in symbolic form, which congruence do you use? DoesÂ âˆ ABD equalÂ âˆ ACD? Why or why not?

Solution:

Yes, the two triangles are congruent because given that ABC and DBC have side BC common, AB = BD and AC = CD

Also from the above data we can say

By SSS criterion of congruency,Â Î”ABCÂ â‰…Â Î”DBC

No,Â âˆ ABD andÂ âˆ ACD are not equal because ABÂ not equal toÂ AC